Oscillatory surface rheotaxis of swimming E. coli bacteria

Abstract

Bacterial contamination of biological channels, catheters or water resources is a major threat to public health, which can be amplified by the ability of bacteria to swim upstream. The mechanisms of this 'rheotaxis', the reorientation with respect to flow gradients, are still poorly understood. Here, we follow individual E. coli bacteria swimming at surfaces under shear flow using 3D Lagrangian tracking and fluorescent flagellar labelling. Three transitions are identified with increasing shear rate: Above a first critical shear rate, bacteria shift to swimming upstream. After a second threshold, we report the discovery of an oscillatory rheotaxis. Beyond a third transition, we further observe coexistence of rheotaxis along the positive and negative vorticity directions. A theoretical analysis explains these rheotaxis regimes and predicts the corresponding critical shear rates. Our results shed light on bacterial transport and reveal strategies for contamination prevention, rheotactic cell sorting, and microswimmer navigation in complex flow environments.

Fig. 1

Experimental observations of oscillatory rheotaxis. a Set-up geometry. b Various types of surface trajectories obtained from 3D tracking at shear rates $\dot{\gamma }=1-50{\mathrm{s}}^{-1}$ (colours), shown in the lab frame and arranged according to increasing shear. Circles indicate the initial positions. c–f Oscillatory rheotaxis. c Typical temporal evolution of the transverse velocity v_z(t) from a 3D tracking experiment that oscillates to the left (marked with an asterisk in panel b), at $\dot{\gamma }=49{\mathrm{s}}^{-1}$. The uncertainty Δv_z ≈ 1.5 μm s⁻¹ (error bars; top right). d Typical temporal evolution of the in-plane angle ψ(t) from a fluorescence experiment at $\dot{\gamma }=32{\mathrm{s}}^{-1}$. The uncertainty Δψ ≈ 5.5° (error bars; top right). e Time lapse of an oscillating bacterium with fluorescently stained flagella, using 10 fps snapshots overlaid to highlight its trajectory, taken in the Lagrangian reference frame of the average downstream bacterial velocity. f Oscillation frequency versus shear rate, obtained from Fourier transformation of v_z(t) in 3D tracking experiments (green hexagons), of ψ(t) in fluorescence experiments (magenta stars), and of ψ(t) in simulations (blue triangles). Horizontal error bars stem from the uncertainty of the bacterial z position, and vertical error bars correspond to two standard deviations from the ensemble mean, the 95% confidence interval. Ninety-four trajectories from 3D tracking experiments and 34 trajectories from fluorescence experiments, corresponding each to a different bacteria, have been analysed. Overlaid are theoretical estimates for the oscillation frequency (Eq. (23); dashed yellow line) and the circling frequency (ν_C/2π; dotted blue line). Parameters used are: ν_W = 4 s⁻¹, θ₀ = −10°, ν_C = 1 s⁻¹, Γ = 4, ${\overline{\nu }}_{\mathrm{H}}=0.02$, ${\overline{\nu }}_{\mathrm{V}}=0.5$, θ_V = 2.3°, θ_E = 20°, v_s = 20 μm s⁻¹, h_s = 1 μm, D_r = 0.057 s⁻¹

Fig. 2

Summary of reorientation mechanisms included in the model. Wall effects: a Steric and hydrodynamic interactions align swimmers with surfaces. b Clock-wise torque from the counter-rotation of the cell body and flagella. Flow effects: c Left-handed helical flagellar bundle in shear reorients swimmers to the right. d Jeffery orbits of elongated bacteria. Flow-wall coupling: e Weathervane effect reorients swimmers to the upstream direction. For all these individual contributions (a–e), the corresponding orientation dynamics in ψ–θ phase space are shown in Supplementary Fig. 1. Combinations of the different effects give f swimming in the upstream direction (a, b, e), g bulk reorientation, biased to the right due to flagellar chirality (c, d), and h oscillatory swimming, oriented slightly upstream due to the weathervane effect (d, e). Green (red) stars are stable (unstable) orientation fixed points, and the blue diamond is a saddle point. The parameters used are given in the caption of Fig. 1, with shear rate $\dot{\gamma }=5{\mathrm{s}}^{-1}$

Fig. 3

Characterisation of the four different surface rheotaxis regimes. Shown are simulated trajectories in the laboratory frame (upper panels) and the corresponding orientation dynamics (lower panels) with increasing shear rate: a (I, $\dot{\gamma }=1{\mathrm{s}}^{-1}$) Circular swimming with a bias to the right. b (II, $\dot{\gamma }=5{\mathrm{s}}^{-1}$) Upstream motion. c (III, $\dot{\gamma }=25{\mathrm{s}}^{-1}$) Oscillatory motion, increasingly more to the right. d (IV, $\dot{\gamma }=60{\mathrm{s}}^{-1}$) Coexistence between swimming to the right and to the left, with dynamical switching between these. Black circles indicate the initial swimmer positions. The parameters used are given in the caption of Fig. 1

Fig. 4

Sketch of the oscillatory rheotaxis mechanism. Here the bacterium is initially oriented towards the right and slightly downstream, and red arrows show the projection of the cell onto the surface. Then, the oscillations can be envisaged as a 4-step process: a The vorticity pushes the body down onto the surface and lifts the flagella up. b Then the flow advects the flagella faster than the body, rotating the bacterium about the y axis to the upstream direction. The weathervane effect enhances this rotation as the cell pivots about the anchoring point. c Now the vorticity pushes the flagella onto the wall and lifts the body up. d Subsequently the body is advected faster, rotating the swimmer back to the downstream direction. This cycle is repeated, leading to oscillatory trajectories. Note that this is a simplified picture and all surface and flow effects (Fig. 2) contribute to the dynamics at any one time

Fig. 5

Bacterial orientation as a function of applied shear. a, b Distributions of the in-plane angle ψ and the pitch angle θ, respectively, obtained by simulating N = 10⁴ trajectories over long times until steady state is reached, for different shear rates $\dot{\gamma }$. The parameters used are the same as in Fig. 1. Inset: same on a logarithmic scale. c, d Rheotaxis phase diagrams. Shown are the equilibrium in-plane angle ψ* and the equilibrium pitch angle θ*, respectively, obtained numerically from the full deterministic model (solid lines), analytically (dashed lines), from simulations (points) and from experiments (magenta stars). The four regimes (blue and white areas) are separated by critical shear rates ${\dot{\gamma }}_c$